FIGURE AND STABILITY OF A LIQUID SATELLITE. 
211 
I find finally that the equation of condition is /( “) = 0, where 
/ (;) = 
9 l + A 1/3 + A 2/3 a 3a (AQ-'F) 
2 (1 + A) 13 (1 + A 13 ) r 2(1 + A) 1 ' 3 (1 - A 2 ' 3 ) 
1 r _fi£l{48-20A-(l9-15A) cos 2 y- (9-25A) cos 2 (3} 
(1 + A) 1/3 (1 — A 2 ' 3 ) 
a n- 
+ —^ 3 {20 —48A— (15 —19A) cos 2 Y- (25-9A) cos 2 B} 
2 Or 
Q n 
+ ^_{288-112A-(260-140A)cos 2 y-(204-196A) cos 2 (3 
+ (87-63A) cos 4 y + (59-91A) cos 4 /3 
+ (30 — 70A) cos 2 (3 cos 2 y} 
+ {ll2-288A-(140-260A)cos 2 r —(196 —204A)cos 2 B 
+ (63-87A) cos 4 T+ (91 -59A) cos 4 B 
+ (70 —30A) cos 2 B cos 2 T} 
+ f {40(1 -A) - (18-22A) cos 2 y — (22-18A) cos 2 T 
- (14-2GA) cos 2 /3 — (2G-14A) cos 2 B 
+15 (1 — A) (cos 2 y cos 2 r+ cos 2 (3 cos 2 B) 
+ (7 —3A)cos 2 y cos 2 B + (3 —7A)cos 2 T cos 2 /?} (44). 
In this expression c and C are respectively the longest semi-axes of the two 
ellipsoids, which are pointed at one another. 
We may derive ifj and 'F from Legendre’s tables of elliptic integrals for 
^ • U«,r). * = d kr^r); 
c sin 
r 
or we may expand the integrals in powers of k' 2 tan 2 y and obtain the approximate 
formula 
^ = ;[i + (n-i)(i+}«' 3 + i V‘+A^«'*-)-U 2 tau 2 r (i+A K ' !! +- 1 % K '*...) 
+ ^K n tan 4 y(l + 3 ^K /2 ...)-/ ( .V 6 tan 6 y (1 + ...)] . (45), 
where Cl = — ^ log c ^ Sm — • The formula for 'F is of course symmetrical, 
sm y cos y 
It should be noted that when the two ellipsoids reduce to spheres, we have 
y = /3 = r= B = 0, c = 
^ : ' a Q - 1 a xb = - = - ! ^F = - (1 +A) 1/3 
(1+A) 1/3 ’ (1+A) 1 ^ 3 ’ ^ c a A 1/3 ’ a 1 '' 
2 e 2 
